Analysis of fuzzified boundary value problems for MHD Couette and Poiseuille flow

In an uncertain atmosphere, the magnetohydrodynamics (MHD) flow in three principal flows of the third grade fluid across two parallel plates is presented. Fuzzy differential equations are constructed by manipulating dimensionless differential equations. The prime purpose of the current article is to use a semi-analytical approach fuzzy-based Adomian decomposition method to achieve numerical results for nonlinear FDEs with fuzzy boundary conditions. Triangular fuzzy numbers are used in fuzzy BCs with help of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha {\text{-cut}}$$\end{document}α-cut approach. This strategy is linked to the membership function. In a graphic and tabular depiction, the effect of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α and other constraints on fuzzy velocity profiles is explored. The current findings are in good agreement with their previous numerical and analytical results in a crisp environment.

Scientists are paying close attention to non-Newtonian fluids because of their frequent practices in the industry, science, and, engineering such as mayonnaise, soap, cosmetics, paints, biological solutions, blood, shampoos, glues, tars, syrups, yoghurt, and other industrial materials fall into this category. As a result, researchers have given differential type 1 fluids a higher priority. We will focus on well known third grade fluid or differential types, which have been extensively researched in a variety of flow processes [2][3][4][5][6][7][8][9] . The study of three basic streams (especially, Poiseuille, Couette, and generalized Couette flow) attracts investigators across several non-Newtonian fluids owing to their potential applications in industries and science. Injection moulding, continuous casting, die flow, plastic forming, extrusion, and asthenosphere flows are examples of unidirectional flows utilised in polymer engineering [10][11][12][13] . The study of electrically conducting liquids moving in the presence of a magnetic field is known as MHD. MHD flow across infinite parallel plates has important applications in geophysical, geothermal reservoirs, metallurgical processing, mineral industries, pumps, astrophysical, MHD generators, polymer technology, and other fields. MHD liquid is a lubricant that prevents lubricant viscosity from changing suddenly with temperature in industrial and other applications. Kamran and Siddique 14 investigated the three flow problems (Poiseuille, Couette, and generalized Couette flow) on MHD third grade fluid across the two parallel plates with help of ADM. There is a lot of literature on this topic, such as 11,[15][16][17][18] .
The ADM was first proposed by Adomian [19][20][21] . ADM is a procedure for solving linear and nonlinear (DEs) that is both trustworthy and efficient. The ADM has several benefits over other analytical and numerical approaches, notably the absence of perturbation, linearization, discretization, or spatial translation. ADM was utilised by Siddiqui et al. 22 to examine the parallel plates flow of a third grade fluid and the results were compared to numerical methods. Pirzaada and Vakaaskar 23 used fuzzy ADM to find the solution to the fuzzy heat equation. Paripour et al. 24 evaluated the fuzzy ADM and predictor-corrector (PC) strategies for the numerical solutions of hybrid FDEs, concluding that the ADM is superior to the PC method. In addition, For squeezed flow between the two circular plates, Siddiquie et al. 25 compared the ADM to the homotopy perturbation technique (HPM). ADM outperforms HPM, as per their observations. Biswal et al. 26 investigated the spontaneous convection of nanofluid flow over two parallel plates by HPM in an uncertain atmosphere. TFN stands for nanoparticle volume percentage, as well as the fact that a fuzzy output is preferable to a crisp one.
In science and engineering, fluid flow is extremely important. An increase in a wide variety of issues such as magnetic effect, chemical diffusion, and heat transfer. These physical problems are then transformed into linear or nonlinear DEs after being governed. The solution of DEs is highly influenced by physical difficulties otherwise. The TFNs with peak (center) χ, right width η − χ > 0, left width χ − δ > 0, and these TFNs are transformed into interval numbers through α-cut approach, is written as where 0 ≤ α ≤ 1. TFNs satisfy the following conditions: (1) 1]. (iv) v(x; α) and u(x; α) are bounded on left continuous and right continuous at [0, 1] respectively. " Definition 28 Let I ∓ be an interval and I ∓ ⊆ R. A mapping u : I ∓ → F * is called a fuzzy process, defined as Definition 28 Let I ∓ ⊆ R and u(x) be a fuzzy valued function define on I ∓ . Let u(x; α) = [v(x; α), u(x; α)] for all α-cut. Assume that v(x; α) and u(x; α) have continuous derivatives or differentiable, for all x ∈ I ∓ and . Similarly, we can define higher-order derivatives in the same way. Then du(x; α) dx , satisfy the following conditions:

Basic equations
The following equations describe the flow of an incompressible unidirectional third-grade fluid with MHD effects: where the density (ρ), stress tensor (τ * * ), pressure p , velocity vector (V), viscosity (µ), electric current (J), and total magnetic field (B). B = B • + b , where induced magnetic field (b) and imposed magnetic field (B o ). The modified Ohm's law and Maxwell's equations 14 are applicable in the absence of displacement currents.
Define velocity profile for one-dimensional flows as: and modified pressure p ∓ is For simplicity, the momentum Eq. (2) along with Eqs. where L 1 , q, and N 1 are linear, source term, and non-linear operators respectively. Also, the operator L 1 can be written as here, L is the highest order derivative in L 1 and R 1 is the remaining operator in L 1 whose order is less than the order of L . From (12) and (13) we have where g(x; α) signifies the terms arising after integration of q(x; α) and calculate constants of integration with the help of boundary conditions. So, u ∓ (x; α) and N 1 u ∓ (x; α) can be written as 19-21 , where A * n, s are called Adomian polynomials 19,20 . The algorithm of the general ADM can be communicated as  plate at x = d is moving with constant velocity U while The lower plate is fixed. The magnetic field is applied vertically upward in a non-conducting manner to both plates. Also, assume that the normal flow is in y-direction while the x-axis is engaged as the way of flow (see in Fig. 1 To deal with these problems, we used TFNs (δ, χ, η) and (d, e, f ). Because the boundary of the parallel plates is treated as fuzzified, this discretization is applied in the boundary for certain flow behaviour. The Eqs. (24 and 25) are converted to an FDE is given below The fuzzy BCs become Solving Eqs. (35) to (39) and putting all values of v 0 (x; α), v 1 (x; α), . . . in Eq. (40) we have Using the above, the solution of v(x, α) and u(x, α) are The Solutions of v(x, α) and u(x, α) are

Results and discussion
In a fuzzy environment, discuss the three elementary flow problems of a third grade fluid such as plane Poiseuille, Couette, and generalised Couette flow. The governing equations convert into FDEs for the analytical solutions using fuzzy ADM to find the fuzzy velocity profiles of a third-grade differential type fluid among two parallel plates with MHD (magnetic parameter m) effect under a constant pressure gradient dp/dy = p . Figures 3, 4 4 and Yürüsoy 9 . The validation of the present study findings was determined to be in excellent agreement.           Figure 11 shows the v(x, α) and u(x, α) for various α values. Because the crisp velocity profile lies between the v(x, α) and u(x, α) , the fuzzy velocity drops into the crisp velocity profile when α = 1, indicating that the current problem is a expansion of Kamran and Siddique 14 . Figures 12 and 13 depict the uncertain response of the TFN memberships function with the triangle fuzzy plot when β and m are varied. Figure 12 shows the fuzzy width declines through growing input m, but Fig. 13 demonstrates how the uncertain                     Table 2. Comparison of analytical results for the crisp velocity profile of Poiseuille flow when β = 0.1, m = 0, p = −0.5 and α-cut = 1.
x Siddiqui et al. 4 Yürüsoy et al. 9 Kamran and Siddique 14    x Siddiqui et al. 4 Yürüsoy et al. 9 Kamran and Siddique 14 Table 4.  Fig. 27, the uncertain width suddenly grows when the value of p is increased. It was also discovered that when α grows, the v(x, α) increases and the higher drops, implying that the solutions are strong. The breadth between the v(x, α) and u(x, α) narrows as α grows, and at α = 1, they coherent with the traditional solution. The evaluation of v(x, α) , mid, and u(x, α) at various x values using p = -0.2, β = 0.3, and m = 0.2 are shown in Table 5. Furthermore, every fixed α-cut, fuzzy velocity profiles always shift within a particular range, and the range steadily declines as α-cut values increase.  Fig. 43, the fuzzy width quickly rises as p is enhanced. Table 6 presents the assessment of v(x, α) , mid, and u(x, α) velocity profiles at various x values using p = 0.015, m = 0.1 and β = 0.2, as fixed values. Moreover, each fixed α-cut, fuzzy velocity profile always shift within a particular range, and the range steadily reduces as α values increase. The solutions are well-suited in the aforementioned discussions; the crisp solution is sandwiched between the fuzzy solutions (lower and upper-velocity profiles), and α approaching one position the fuzzy solutions are close to the crisp solution. The fuzzy velocity profile of the fluid is a better choice than the crisp or classical velocity profile of the fluid, according to the conclusion of the entire discussion. The single flow situation of fluid is represented by a crisp or classical velocity profile, but the interval flow situation is represented by a fuzzy velocity profile, which has lower and higher boundaries. In addition, the model described a new feature at various α values and gave accurate solution intervals (lower and upper-velocity profiles) for better dynamic analysis judgment. Table 6. Fuzzy solution of v(x; α) , mid and u(x; α) at α-cut = 0, p = 0.015, β = 0.2, and m = 0.01, with varous values of x. www.nature.com/scientificreports/ Figures 3,4,5,6,7,8,9,10,11,12,13 and Table 4. Figures 14,15,16,17,18,19,20,21,22,23,24,25,26,27 and Table 5.

Conclusions
The three fundamental flow phenomena that inevitably arise in the study of fluid dynamics, especially plane Poiseuille, plane Couette, and generalised Couette flow of a non-Newtonian fluid under the impact of MHD force in a fuzzy environment, have been investigated in this work. The dimensionless governing DEs are discretized into FDEs with fuzzified BCs, and ADM is used to resolve them. When compared to previous results, the current crisp results acquired by ADM are shown to be in excellent agreement. The TFNs are utilised for uncertainties on the dynamic behaviour of the said problem. The velocity profiles (lower and higher) grow when the β and α-cut increase, whereas the fuzzy velocity profile decrease as the m increases in three flow situations. The range of predicted lower and upper-velocity profiles is dependent on the α-cut, according to the findings. The end outcome is always an envelope of solutions with a crisp solution in the middle. As a result, As a result, fuzzy velocity fields are the modification of the crisp velocity field of a third grade fluid flowing between two parallel plates.

Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.